NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 47 - 54 of 101

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Question 47

Appeared in Year: 2009

Write in Short

Short Answer▾

Let

Write down the residues of at each of its poles

Question 48

Appeared in Year: 2007

Describe in Detail

Essay▾

Let C denote the Boundary of the square whose sides are given by the lines . Assume that C is described in the position sense, i.e.. anticlockwise. Evaluate

Explanation

Boundary of the Square

To evaluate

Take

Here is a pole of order 1 and lies within the square.

These are simple poles which lie outside the square

So they will not contribute towards the residue of

By Cauchy residue theorem

Question 49

Appeared in Year: 2014

Describe in Detail

Essay▾

Let be differentiable at and let . Evaluate

Explanation

Now

So this is form (Exponential form)

Now using the result

Or if

Then

… eq. (1)

form

From (1)

… (1 more words) …

Question 50

Appeared in Year: 2009

Describe in Detail

Essay▾

Let C be the contour consisting of the lines , described counterclockwise in the plane. Compute

Explanation

is the contour shown in the figure

The Contour Consisting of the Lines

To evaluate

Now is a pole of order 1 (simple pole)

Also lies within C

=

By Cauchy residue theorem

Question 51

Appeared in Year: 2017

Describe in Detail

Essay▾

Let Write down the Taylor Expansion of f in the neighbourhood of .

Explanation

Here

Put

This is the required Taylor expansion of f in the neighbourhood of

Question 52

Appeared in Year: 2015

Describe in Detail

Essay▾

In each of the following cases, state whether the given series is absolutely convergent, conditionally convergent or divergent

a)

b)

c)

Explanation

Option (a) The given series is

Where

Now comparing it with

1)

2)

3)

By Leibnitz test (alternating series test)

The given series is convergent

Which is finite & non – zero

converge or diverge together

But is divergent by p – test

is divergent (By comparison test)

is not absolutely convergent

is conditionally convergent

Option (b) The given series i…

… (44 more words) …

Question 53

Appeared in Year: 2006

Question

MCQ▾

Pick out series which are absolutely convergent.

Choices

Choice (4)Response

a.

b.

c.

d.

Both a. and b. are correct

Question 54

Appeared in Year: 2005

Describe in Detail

Essay▾

What are the value of for which the following series is convergent?

Explanation

Let the omen series be denoted by … eq. (1)

Where

Since for absolute convergence, we take

Now is convergent if … eq. (2)

And divergent if (By p-test)

Also eq. (1) is convergent for by Leibnitz test

is convergent by Leibnitz test for … eq. (3)

From eq. (2) & (3)

We get eq. (1) is convergent

… (3 more words) …